ĐKXĐ: \(x>0;x\ne1\)
\(K=\left(\frac{x+\sqrt{x}+1}{x+1}\right):\left(\frac{x+1}{\left(x+1\right)\left(\sqrt{x}-1\right)}-\frac{2\sqrt{x}}{\left(x+1\right)\left(\sqrt{x}-1\right)}\right)\)
\(=\left(\frac{x+\sqrt{x}+1}{x+1}\right):\left(\frac{x-2\sqrt{x}+1}{\left(x+1\right)\left(\sqrt{x}-1\right)}\right)=\frac{\left(x+\sqrt{x}+1\right)\left(x+1\right)\left(\sqrt{x}-1\right)}{\left(x+1\right)\left(\sqrt{x}-1\right)^2}\)
\(=\frac{x+\sqrt{x}+1}{\sqrt{x}-1}\)
\(x=4+2\sqrt{3}=\left(\sqrt{3}+1\right)^2\Rightarrow x=\sqrt{3}+1\)
\(\Rightarrow K=\frac{4+2\sqrt{3}+\sqrt{3}+1+1}{\sqrt{3}+1-1}=\frac{6+3\sqrt{3}}{\sqrt{3}}=3+2\sqrt{3}\)
\(K>1\Leftrightarrow\frac{x+\sqrt{x}+1}{\sqrt{x}-1}>1\Leftrightarrow\frac{x+2}{\sqrt{x}-1}>0\)
\(\Leftrightarrow\sqrt{x}-1>0\Leftrightarrow x>1\)
